The Maseeh Mathematics & Statistics
Colloquium Series.

Abstract: The octonions are the last in the sequence of 4 division algebras generalizing the real and complex numbers. The octonions play a somewhat surprising role in a wide range of geometric phenomena, some of which will be described in this talk. For instance, the octonions provide a natural description of the rotation groups SO(7) and SO(8), and explain why the vector cross product only exists in 3 and 7 dimensions, while 2x2 octonionic Hermitian matrices provide a natural description of the Lorentz group SO(9,1) (and the last Hopf fibration), and explain why supersymmetry only exists in 3, 4, 6, and 10 dimensions. Furthermore, the exceptional groups G2, F4, and E6 have natural octonionic descriptions, with applications to both quantum mechanics and particle physics.